/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.itrs /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- YES proof of /export/starexec/sandbox2/benchmark/theBenchmark.itrs # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty Termination of the given ITRS could be proven: (0) ITRS (1) ITRStoIDPProof [EQUIVALENT, 0 ms] (2) IDP (3) UsableRulesProof [EQUIVALENT, 0 ms] (4) IDP (5) IDPNonInfProof [SOUND, 268 ms] (6) IDP (7) IDPNonInfProof [SOUND, 2109 ms] (8) IDP (9) IDependencyGraphProof [EQUIVALENT, 0 ms] (10) TRUE ---------------------------------------- (0) Obligation: ITRS problem: The following function symbols are pre-defined: <<< & ~ Bwand: (Integer, Integer) -> Integer >= ~ Ge: (Integer, Integer) -> Boolean | ~ Bwor: (Integer, Integer) -> Integer / ~ Div: (Integer, Integer) -> Integer != ~ Neq: (Integer, Integer) -> Boolean && ~ Land: (Boolean, Boolean) -> Boolean ! ~ Lnot: (Boolean) -> Boolean = ~ Eq: (Integer, Integer) -> Boolean <= ~ Le: (Integer, Integer) -> Boolean ^ ~ Bwxor: (Integer, Integer) -> Integer % ~ Mod: (Integer, Integer) -> Integer + ~ Add: (Integer, Integer) -> Integer > ~ Gt: (Integer, Integer) -> Boolean -1 ~ UnaryMinus: (Integer) -> Integer < ~ Lt: (Integer, Integer) -> Boolean || ~ Lor: (Boolean, Boolean) -> Boolean - ~ Sub: (Integer, Integer) -> Integer ~ ~ Bwnot: (Integer) -> Integer * ~ Mul: (Integer, Integer) -> Integer >>> The TRS R consists of the following rules: eval(x, y) -> Cond_eval(x > y, x, y) Cond_eval(TRUE, x, y) -> eval(x, y + 1) eval(x, y) -> Cond_eval1(y > x && x > y, x, y) Cond_eval1(TRUE, x, y) -> eval(x, y + 1) eval(x, y) -> Cond_eval2(x > y && y >= x, x, y) Cond_eval2(TRUE, x, y) -> eval(x + 1, y) eval(x, y) -> Cond_eval3(y > x && y >= x, x, y) Cond_eval3(TRUE, x, y) -> eval(x + 1, y) The set Q consists of the following terms: eval(x0, x1) Cond_eval(TRUE, x0, x1) Cond_eval1(TRUE, x0, x1) Cond_eval2(TRUE, x0, x1) Cond_eval3(TRUE, x0, x1) ---------------------------------------- (1) ITRStoIDPProof (EQUIVALENT) Added dependency pairs ---------------------------------------- (2) Obligation: IDP problem: The following function symbols are pre-defined: <<< & ~ Bwand: (Integer, Integer) -> Integer >= ~ Ge: (Integer, Integer) -> Boolean | ~ Bwor: (Integer, Integer) -> Integer / ~ Div: (Integer, Integer) -> Integer != ~ Neq: (Integer, Integer) -> Boolean && ~ Land: (Boolean, Boolean) -> Boolean ! ~ Lnot: (Boolean) -> Boolean = ~ Eq: (Integer, Integer) -> Boolean <= ~ Le: (Integer, Integer) -> Boolean ^ ~ Bwxor: (Integer, Integer) -> Integer % ~ Mod: (Integer, Integer) -> Integer + ~ Add: (Integer, Integer) -> Integer > ~ Gt: (Integer, Integer) -> Boolean -1 ~ UnaryMinus: (Integer) -> Integer < ~ Lt: (Integer, Integer) -> Boolean || ~ Lor: (Boolean, Boolean) -> Boolean - ~ Sub: (Integer, Integer) -> Integer ~ ~ Bwnot: (Integer) -> Integer * ~ Mul: (Integer, Integer) -> Integer >>> The following domains are used: Integer, Boolean The ITRS R consists of the following rules: eval(x, y) -> Cond_eval(x > y, x, y) Cond_eval(TRUE, x, y) -> eval(x, y + 1) eval(x, y) -> Cond_eval1(y > x && x > y, x, y) Cond_eval1(TRUE, x, y) -> eval(x, y + 1) eval(x, y) -> Cond_eval2(x > y && y >= x, x, y) Cond_eval2(TRUE, x, y) -> eval(x + 1, y) eval(x, y) -> Cond_eval3(y > x && y >= x, x, y) Cond_eval3(TRUE, x, y) -> eval(x + 1, y) The integer pair graph contains the following rules and edges: (0): EVAL(x[0], y[0]) -> COND_EVAL(x[0] > y[0], x[0], y[0]) (1): COND_EVAL(TRUE, x[1], y[1]) -> EVAL(x[1], y[1] + 1) (2): EVAL(x[2], y[2]) -> COND_EVAL1(y[2] > x[2] && x[2] > y[2], x[2], y[2]) (3): COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(x[3], y[3] + 1) (4): EVAL(x[4], y[4]) -> COND_EVAL2(x[4] > y[4] && y[4] >= x[4], x[4], y[4]) (5): COND_EVAL2(TRUE, x[5], y[5]) -> EVAL(x[5] + 1, y[5]) (6): EVAL(x[6], y[6]) -> COND_EVAL3(y[6] > x[6] && y[6] >= x[6], x[6], y[6]) (7): COND_EVAL3(TRUE, x[7], y[7]) -> EVAL(x[7] + 1, y[7]) (0) -> (1), if (x[0] > y[0] & x[0] ->^* x[1] & y[0] ->^* y[1]) (1) -> (0), if (x[1] ->^* x[0] & y[1] + 1 ->^* y[0]) (1) -> (2), if (x[1] ->^* x[2] & y[1] + 1 ->^* y[2]) (1) -> (4), if (x[1] ->^* x[4] & y[1] + 1 ->^* y[4]) (1) -> (6), if (x[1] ->^* x[6] & y[1] + 1 ->^* y[6]) (2) -> (3), if (y[2] > x[2] && x[2] > y[2] & x[2] ->^* x[3] & y[2] ->^* y[3]) (3) -> (0), if (x[3] ->^* x[0] & y[3] + 1 ->^* y[0]) (3) -> (2), if (x[3] ->^* x[2] & y[3] + 1 ->^* y[2]) (3) -> (4), if (x[3] ->^* x[4] & y[3] + 1 ->^* y[4]) (3) -> (6), if (x[3] ->^* x[6] & y[3] + 1 ->^* y[6]) (4) -> (5), if (x[4] > y[4] && y[4] >= x[4] & x[4] ->^* x[5] & y[4] ->^* y[5]) (5) -> (0), if (x[5] + 1 ->^* x[0] & y[5] ->^* y[0]) (5) -> (2), if (x[5] + 1 ->^* x[2] & y[5] ->^* y[2]) (5) -> (4), if (x[5] + 1 ->^* x[4] & y[5] ->^* y[4]) (5) -> (6), if (x[5] + 1 ->^* x[6] & y[5] ->^* y[6]) (6) -> (7), if (y[6] > x[6] && y[6] >= x[6] & x[6] ->^* x[7] & y[6] ->^* y[7]) (7) -> (0), if (x[7] + 1 ->^* x[0] & y[7] ->^* y[0]) (7) -> (2), if (x[7] + 1 ->^* x[2] & y[7] ->^* y[2]) (7) -> (4), if (x[7] + 1 ->^* x[4] & y[7] ->^* y[4]) (7) -> (6), if (x[7] + 1 ->^* x[6] & y[7] ->^* y[6]) The set Q consists of the following terms: eval(x0, x1) Cond_eval(TRUE, x0, x1) Cond_eval1(TRUE, x0, x1) Cond_eval2(TRUE, x0, x1) Cond_eval3(TRUE, x0, x1) ---------------------------------------- (3) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (4) Obligation: IDP problem: The following function symbols are pre-defined: <<< & ~ Bwand: (Integer, Integer) -> Integer >= ~ Ge: (Integer, Integer) -> Boolean | ~ Bwor: (Integer, Integer) -> Integer / ~ Div: (Integer, Integer) -> Integer != ~ Neq: (Integer, Integer) -> Boolean && ~ Land: (Boolean, Boolean) -> Boolean ! ~ Lnot: (Boolean) -> Boolean = ~ Eq: (Integer, Integer) -> Boolean <= ~ Le: (Integer, Integer) -> Boolean ^ ~ Bwxor: (Integer, Integer) -> Integer % ~ Mod: (Integer, Integer) -> Integer + ~ Add: (Integer, Integer) -> Integer > ~ Gt: (Integer, Integer) -> Boolean -1 ~ UnaryMinus: (Integer) -> Integer < ~ Lt: (Integer, Integer) -> Boolean || ~ Lor: (Boolean, Boolean) -> Boolean - ~ Sub: (Integer, Integer) -> Integer ~ ~ Bwnot: (Integer) -> Integer * ~ Mul: (Integer, Integer) -> Integer >>> The following domains are used: Integer, Boolean R is empty. The integer pair graph contains the following rules and edges: (0): EVAL(x[0], y[0]) -> COND_EVAL(x[0] > y[0], x[0], y[0]) (1): COND_EVAL(TRUE, x[1], y[1]) -> EVAL(x[1], y[1] + 1) (2): EVAL(x[2], y[2]) -> COND_EVAL1(y[2] > x[2] && x[2] > y[2], x[2], y[2]) (3): COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(x[3], y[3] + 1) (4): EVAL(x[4], y[4]) -> COND_EVAL2(x[4] > y[4] && y[4] >= x[4], x[4], y[4]) (5): COND_EVAL2(TRUE, x[5], y[5]) -> EVAL(x[5] + 1, y[5]) (6): EVAL(x[6], y[6]) -> COND_EVAL3(y[6] > x[6] && y[6] >= x[6], x[6], y[6]) (7): COND_EVAL3(TRUE, x[7], y[7]) -> EVAL(x[7] + 1, y[7]) (0) -> (1), if (x[0] > y[0] & x[0] ->^* x[1] & y[0] ->^* y[1]) (1) -> (0), if (x[1] ->^* x[0] & y[1] + 1 ->^* y[0]) (1) -> (2), if (x[1] ->^* x[2] & y[1] + 1 ->^* y[2]) (1) -> (4), if (x[1] ->^* x[4] & y[1] + 1 ->^* y[4]) (1) -> (6), if (x[1] ->^* x[6] & y[1] + 1 ->^* y[6]) (2) -> (3), if (y[2] > x[2] && x[2] > y[2] & x[2] ->^* x[3] & y[2] ->^* y[3]) (3) -> (0), if (x[3] ->^* x[0] & y[3] + 1 ->^* y[0]) (3) -> (2), if (x[3] ->^* x[2] & y[3] + 1 ->^* y[2]) (3) -> (4), if (x[3] ->^* x[4] & y[3] + 1 ->^* y[4]) (3) -> (6), if (x[3] ->^* x[6] & y[3] + 1 ->^* y[6]) (4) -> (5), if (x[4] > y[4] && y[4] >= x[4] & x[4] ->^* x[5] & y[4] ->^* y[5]) (5) -> (0), if (x[5] + 1 ->^* x[0] & y[5] ->^* y[0]) (5) -> (2), if (x[5] + 1 ->^* x[2] & y[5] ->^* y[2]) (5) -> (4), if (x[5] + 1 ->^* x[4] & y[5] ->^* y[4]) (5) -> (6), if (x[5] + 1 ->^* x[6] & y[5] ->^* y[6]) (6) -> (7), if (y[6] > x[6] && y[6] >= x[6] & x[6] ->^* x[7] & y[6] ->^* y[7]) (7) -> (0), if (x[7] + 1 ->^* x[0] & y[7] ->^* y[0]) (7) -> (2), if (x[7] + 1 ->^* x[2] & y[7] ->^* y[2]) (7) -> (4), if (x[7] + 1 ->^* x[4] & y[7] ->^* y[4]) (7) -> (6), if (x[7] + 1 ->^* x[6] & y[7] ->^* y[6]) The set Q consists of the following terms: eval(x0, x1) Cond_eval(TRUE, x0, x1) Cond_eval1(TRUE, x0, x1) Cond_eval2(TRUE, x0, x1) Cond_eval3(TRUE, x0, x1) ---------------------------------------- (5) IDPNonInfProof (SOUND) Used the following options for this NonInfProof: IDPGPoloSolver: Range: [(-1,2)] IsNat: false Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpDefaultShapeHeuristic@63d55717 Constraint Generator: NonInfConstraintGenerator: PathGenerator: MetricPathGenerator: Max Left Steps: 1 Max Right Steps: 1 The constraints were generated the following way: The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps: Note that final constraints are written in bold face. For Pair EVAL(x, y) -> COND_EVAL(>(x, y), x, y) the following chains were created: *We consider the chain EVAL(x[0], y[0]) -> COND_EVAL(>(x[0], y[0]), x[0], y[0]), COND_EVAL(TRUE, x[1], y[1]) -> EVAL(x[1], +(y[1], 1)) which results in the following constraint: (1) (>(x[0], y[0])=TRUE & x[0]=x[1] & y[0]=y[1] ==> EVAL(x[0], y[0])_>=_NonInfC & EVAL(x[0], y[0])_>=_COND_EVAL(>(x[0], y[0]), x[0], y[0]) & (U^Increasing(COND_EVAL(>(x[0], y[0]), x[0], y[0])), >=)) We simplified constraint (1) using rule (IV) which results in the following new constraint: (2) (>(x[0], y[0])=TRUE ==> EVAL(x[0], y[0])_>=_NonInfC & EVAL(x[0], y[0])_>=_COND_EVAL(>(x[0], y[0]), x[0], y[0]) & (U^Increasing(COND_EVAL(>(x[0], y[0]), x[0], y[0])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (x[0] + [-1] + [-1]y[0] >= 0 ==> (U^Increasing(COND_EVAL(>(x[0], y[0]), x[0], y[0])), >=) & [(-1)bni_25 + (-1)Bound*bni_25] + [(-1)bni_25]y[0] + [(-1)bni_25]x[0] >= 0 & [(-1)bso_26] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (x[0] + [-1] + [-1]y[0] >= 0 ==> (U^Increasing(COND_EVAL(>(x[0], y[0]), x[0], y[0])), >=) & [(-1)bni_25 + (-1)Bound*bni_25] + [(-1)bni_25]y[0] + [(-1)bni_25]x[0] >= 0 & [(-1)bso_26] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (x[0] + [-1] + [-1]y[0] >= 0 ==> (U^Increasing(COND_EVAL(>(x[0], y[0]), x[0], y[0])), >=) & [(-1)bni_25 + (-1)Bound*bni_25] + [(-1)bni_25]y[0] + [(-1)bni_25]x[0] >= 0 & [(-1)bso_26] >= 0) We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (6) (x[0] >= 0 ==> (U^Increasing(COND_EVAL(>(x[0], y[0]), x[0], y[0])), >=) & [(-2)bni_25 + (-1)Bound*bni_25] + [(-2)bni_25]y[0] + [(-1)bni_25]x[0] >= 0 & [(-1)bso_26] >= 0) We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (7) (x[0] >= 0 & y[0] >= 0 ==> (U^Increasing(COND_EVAL(>(x[0], y[0]), x[0], y[0])), >=) & [(-2)bni_25 + (-1)Bound*bni_25] + [(2)bni_25]y[0] + [(-1)bni_25]x[0] >= 0 & [(-1)bso_26] >= 0) (8) (x[0] >= 0 & y[0] >= 0 ==> (U^Increasing(COND_EVAL(>(x[0], y[0]), x[0], y[0])), >=) & [(-2)bni_25 + (-1)Bound*bni_25] + [(-2)bni_25]y[0] + [(-1)bni_25]x[0] >= 0 & [(-1)bso_26] >= 0) For Pair COND_EVAL(TRUE, x, y) -> EVAL(x, +(y, 1)) the following chains were created: *We consider the chain EVAL(x[0], y[0]) -> COND_EVAL(>(x[0], y[0]), x[0], y[0]), COND_EVAL(TRUE, x[1], y[1]) -> EVAL(x[1], +(y[1], 1)) which results in the following constraint: (1) (>(x[0], y[0])=TRUE & x[0]=x[1] & y[0]=y[1] ==> COND_EVAL(TRUE, x[1], y[1])_>=_NonInfC & COND_EVAL(TRUE, x[1], y[1])_>=_EVAL(x[1], +(y[1], 1)) & (U^Increasing(EVAL(x[1], +(y[1], 1))), >=)) We simplified constraint (1) using rule (III) which results in the following new constraint: (2) (>(x[0], y[0])=TRUE ==> COND_EVAL(TRUE, x[0], y[0])_>=_NonInfC & COND_EVAL(TRUE, x[0], y[0])_>=_EVAL(x[0], +(y[0], 1)) & (U^Increasing(EVAL(x[1], +(y[1], 1))), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (x[0] + [-1] + [-1]y[0] >= 0 ==> (U^Increasing(EVAL(x[1], +(y[1], 1))), >=) & [(-1)bni_27 + (-1)Bound*bni_27] + [(-1)bni_27]y[0] + [(-1)bni_27]x[0] >= 0 & [1 + (-1)bso_28] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (x[0] + [-1] + [-1]y[0] >= 0 ==> (U^Increasing(EVAL(x[1], +(y[1], 1))), >=) & [(-1)bni_27 + (-1)Bound*bni_27] + [(-1)bni_27]y[0] + [(-1)bni_27]x[0] >= 0 & [1 + (-1)bso_28] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (x[0] + [-1] + [-1]y[0] >= 0 ==> (U^Increasing(EVAL(x[1], +(y[1], 1))), >=) & [(-1)bni_27 + (-1)Bound*bni_27] + [(-1)bni_27]y[0] + [(-1)bni_27]x[0] >= 0 & [1 + (-1)bso_28] >= 0) We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (6) (x[0] >= 0 ==> (U^Increasing(EVAL(x[1], +(y[1], 1))), >=) & [(-2)bni_27 + (-1)Bound*bni_27] + [(-2)bni_27]y[0] + [(-1)bni_27]x[0] >= 0 & [1 + (-1)bso_28] >= 0) We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (7) (x[0] >= 0 & y[0] >= 0 ==> (U^Increasing(EVAL(x[1], +(y[1], 1))), >=) & [(-2)bni_27 + (-1)Bound*bni_27] + [(-2)bni_27]y[0] + [(-1)bni_27]x[0] >= 0 & [1 + (-1)bso_28] >= 0) (8) (x[0] >= 0 & y[0] >= 0 ==> (U^Increasing(EVAL(x[1], +(y[1], 1))), >=) & [(-2)bni_27 + (-1)Bound*bni_27] + [(2)bni_27]y[0] + [(-1)bni_27]x[0] >= 0 & [1 + (-1)bso_28] >= 0) For Pair EVAL(x, y) -> COND_EVAL1(&&(>(y, x), >(x, y)), x, y) the following chains were created: *We consider the chain EVAL(x[2], y[2]) -> COND_EVAL1(&&(>(y[2], x[2]), >(x[2], y[2])), x[2], y[2]), COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(x[3], +(y[3], 1)) which results in the following constraint: (1) (&&(>(y[2], x[2]), >(x[2], y[2]))=TRUE & x[2]=x[3] & y[2]=y[3] ==> EVAL(x[2], y[2])_>=_NonInfC & EVAL(x[2], y[2])_>=_COND_EVAL1(&&(>(y[2], x[2]), >(x[2], y[2])), x[2], y[2]) & (U^Increasing(COND_EVAL1(&&(>(y[2], x[2]), >(x[2], y[2])), x[2], y[2])), >=)) We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>(y[2], x[2])=TRUE & >(x[2], y[2])=TRUE ==> EVAL(x[2], y[2])_>=_NonInfC & EVAL(x[2], y[2])_>=_COND_EVAL1(&&(>(y[2], x[2]), >(x[2], y[2])), x[2], y[2]) & (U^Increasing(COND_EVAL1(&&(>(y[2], x[2]), >(x[2], y[2])), x[2], y[2])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (y[2] + [-1] + [-1]x[2] >= 0 & x[2] + [-1] + [-1]y[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(>(y[2], x[2]), >(x[2], y[2])), x[2], y[2])), >=) & [(-1)bni_29 + (-1)Bound*bni_29] + [(-1)bni_29]y[2] + [(-1)bni_29]x[2] >= 0 & [(-1)bso_30] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (y[2] + [-1] + [-1]x[2] >= 0 & x[2] + [-1] + [-1]y[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(>(y[2], x[2]), >(x[2], y[2])), x[2], y[2])), >=) & [(-1)bni_29 + (-1)Bound*bni_29] + [(-1)bni_29]y[2] + [(-1)bni_29]x[2] >= 0 & [(-1)bso_30] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (y[2] + [-1] + [-1]x[2] >= 0 & x[2] + [-1] + [-1]y[2] >= 0 ==> (U^Increasing(COND_EVAL1(&&(>(y[2], x[2]), >(x[2], y[2])), x[2], y[2])), >=) & [(-1)bni_29 + (-1)Bound*bni_29] + [(-1)bni_29]y[2] + [(-1)bni_29]x[2] >= 0 & [(-1)bso_30] >= 0) We solved constraint (5) using rule (IDP_SMT_SPLIT). For Pair COND_EVAL1(TRUE, x, y) -> EVAL(x, +(y, 1)) the following chains were created: *We consider the chain EVAL(x[2], y[2]) -> COND_EVAL1(&&(>(y[2], x[2]), >(x[2], y[2])), x[2], y[2]), COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(x[3], +(y[3], 1)) which results in the following constraint: (1) (&&(>(y[2], x[2]), >(x[2], y[2]))=TRUE & x[2]=x[3] & y[2]=y[3] ==> COND_EVAL1(TRUE, x[3], y[3])_>=_NonInfC & COND_EVAL1(TRUE, x[3], y[3])_>=_EVAL(x[3], +(y[3], 1)) & (U^Increasing(EVAL(x[3], +(y[3], 1))), >=)) We simplified constraint (1) using rules (III), (IDP_BOOLEAN) which results in the following new constraint: (2) (>(y[2], x[2])=TRUE & >(x[2], y[2])=TRUE ==> COND_EVAL1(TRUE, x[2], y[2])_>=_NonInfC & COND_EVAL1(TRUE, x[2], y[2])_>=_EVAL(x[2], +(y[2], 1)) & (U^Increasing(EVAL(x[3], +(y[3], 1))), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (y[2] + [-1] + [-1]x[2] >= 0 & x[2] + [-1] + [-1]y[2] >= 0 ==> (U^Increasing(EVAL(x[3], +(y[3], 1))), >=) & [(-1)bni_31 + (-1)Bound*bni_31] + [(-1)bni_31]y[2] + [(-1)bni_31]x[2] >= 0 & [1 + (-1)bso_32] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (y[2] + [-1] + [-1]x[2] >= 0 & x[2] + [-1] + [-1]y[2] >= 0 ==> (U^Increasing(EVAL(x[3], +(y[3], 1))), >=) & [(-1)bni_31 + (-1)Bound*bni_31] + [(-1)bni_31]y[2] + [(-1)bni_31]x[2] >= 0 & [1 + (-1)bso_32] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (y[2] + [-1] + [-1]x[2] >= 0 & x[2] + [-1] + [-1]y[2] >= 0 ==> (U^Increasing(EVAL(x[3], +(y[3], 1))), >=) & [(-1)bni_31 + (-1)Bound*bni_31] + [(-1)bni_31]y[2] + [(-1)bni_31]x[2] >= 0 & [1 + (-1)bso_32] >= 0) We solved constraint (5) using rule (IDP_SMT_SPLIT). For Pair EVAL(x, y) -> COND_EVAL2(&&(>(x, y), >=(y, x)), x, y) the following chains were created: *We consider the chain EVAL(x[4], y[4]) -> COND_EVAL2(&&(>(x[4], y[4]), >=(y[4], x[4])), x[4], y[4]), COND_EVAL2(TRUE, x[5], y[5]) -> EVAL(+(x[5], 1), y[5]) which results in the following constraint: (1) (&&(>(x[4], y[4]), >=(y[4], x[4]))=TRUE & x[4]=x[5] & y[4]=y[5] ==> EVAL(x[4], y[4])_>=_NonInfC & EVAL(x[4], y[4])_>=_COND_EVAL2(&&(>(x[4], y[4]), >=(y[4], x[4])), x[4], y[4]) & (U^Increasing(COND_EVAL2(&&(>(x[4], y[4]), >=(y[4], x[4])), x[4], y[4])), >=)) We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>(x[4], y[4])=TRUE & >=(y[4], x[4])=TRUE ==> EVAL(x[4], y[4])_>=_NonInfC & EVAL(x[4], y[4])_>=_COND_EVAL2(&&(>(x[4], y[4]), >=(y[4], x[4])), x[4], y[4]) & (U^Increasing(COND_EVAL2(&&(>(x[4], y[4]), >=(y[4], x[4])), x[4], y[4])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (x[4] + [-1] + [-1]y[4] >= 0 & y[4] + [-1]x[4] >= 0 ==> (U^Increasing(COND_EVAL2(&&(>(x[4], y[4]), >=(y[4], x[4])), x[4], y[4])), >=) & [(-1)bni_33 + (-1)Bound*bni_33] + [(-1)bni_33]y[4] + [(-1)bni_33]x[4] >= 0 & [(-1)bso_34] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (x[4] + [-1] + [-1]y[4] >= 0 & y[4] + [-1]x[4] >= 0 ==> (U^Increasing(COND_EVAL2(&&(>(x[4], y[4]), >=(y[4], x[4])), x[4], y[4])), >=) & [(-1)bni_33 + (-1)Bound*bni_33] + [(-1)bni_33]y[4] + [(-1)bni_33]x[4] >= 0 & [(-1)bso_34] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (x[4] + [-1] + [-1]y[4] >= 0 & y[4] + [-1]x[4] >= 0 ==> (U^Increasing(COND_EVAL2(&&(>(x[4], y[4]), >=(y[4], x[4])), x[4], y[4])), >=) & [(-1)bni_33 + (-1)Bound*bni_33] + [(-1)bni_33]y[4] + [(-1)bni_33]x[4] >= 0 & [(-1)bso_34] >= 0) We solved constraint (5) using rule (IDP_SMT_SPLIT). For Pair COND_EVAL2(TRUE, x, y) -> EVAL(+(x, 1), y) the following chains were created: *We consider the chain EVAL(x[4], y[4]) -> COND_EVAL2(&&(>(x[4], y[4]), >=(y[4], x[4])), x[4], y[4]), COND_EVAL2(TRUE, x[5], y[5]) -> EVAL(+(x[5], 1), y[5]) which results in the following constraint: (1) (&&(>(x[4], y[4]), >=(y[4], x[4]))=TRUE & x[4]=x[5] & y[4]=y[5] ==> COND_EVAL2(TRUE, x[5], y[5])_>=_NonInfC & COND_EVAL2(TRUE, x[5], y[5])_>=_EVAL(+(x[5], 1), y[5]) & (U^Increasing(EVAL(+(x[5], 1), y[5])), >=)) We simplified constraint (1) using rules (III), (IDP_BOOLEAN) which results in the following new constraint: (2) (>(x[4], y[4])=TRUE & >=(y[4], x[4])=TRUE ==> COND_EVAL2(TRUE, x[4], y[4])_>=_NonInfC & COND_EVAL2(TRUE, x[4], y[4])_>=_EVAL(+(x[4], 1), y[4]) & (U^Increasing(EVAL(+(x[5], 1), y[5])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (x[4] + [-1] + [-1]y[4] >= 0 & y[4] + [-1]x[4] >= 0 ==> (U^Increasing(EVAL(+(x[5], 1), y[5])), >=) & [(-1)bni_35 + (-1)Bound*bni_35] + [(-1)bni_35]y[4] + [(-1)bni_35]x[4] >= 0 & [1 + (-1)bso_36] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (x[4] + [-1] + [-1]y[4] >= 0 & y[4] + [-1]x[4] >= 0 ==> (U^Increasing(EVAL(+(x[5], 1), y[5])), >=) & [(-1)bni_35 + (-1)Bound*bni_35] + [(-1)bni_35]y[4] + [(-1)bni_35]x[4] >= 0 & [1 + (-1)bso_36] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (x[4] + [-1] + [-1]y[4] >= 0 & y[4] + [-1]x[4] >= 0 ==> (U^Increasing(EVAL(+(x[5], 1), y[5])), >=) & [(-1)bni_35 + (-1)Bound*bni_35] + [(-1)bni_35]y[4] + [(-1)bni_35]x[4] >= 0 & [1 + (-1)bso_36] >= 0) We solved constraint (5) using rule (IDP_SMT_SPLIT). For Pair EVAL(x, y) -> COND_EVAL3(&&(>(y, x), >=(y, x)), x, y) the following chains were created: *We consider the chain EVAL(x[6], y[6]) -> COND_EVAL3(&&(>(y[6], x[6]), >=(y[6], x[6])), x[6], y[6]), COND_EVAL3(TRUE, x[7], y[7]) -> EVAL(+(x[7], 1), y[7]) which results in the following constraint: (1) (&&(>(y[6], x[6]), >=(y[6], x[6]))=TRUE & x[6]=x[7] & y[6]=y[7] ==> EVAL(x[6], y[6])_>=_NonInfC & EVAL(x[6], y[6])_>=_COND_EVAL3(&&(>(y[6], x[6]), >=(y[6], x[6])), x[6], y[6]) & (U^Increasing(COND_EVAL3(&&(>(y[6], x[6]), >=(y[6], x[6])), x[6], y[6])), >=)) We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>(y[6], x[6])=TRUE & >=(y[6], x[6])=TRUE ==> EVAL(x[6], y[6])_>=_NonInfC & EVAL(x[6], y[6])_>=_COND_EVAL3(&&(>(y[6], x[6]), >=(y[6], x[6])), x[6], y[6]) & (U^Increasing(COND_EVAL3(&&(>(y[6], x[6]), >=(y[6], x[6])), x[6], y[6])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (y[6] + [-1] + [-1]x[6] >= 0 & y[6] + [-1]x[6] >= 0 ==> (U^Increasing(COND_EVAL3(&&(>(y[6], x[6]), >=(y[6], x[6])), x[6], y[6])), >=) & [(-1)bni_37 + (-1)Bound*bni_37] + [(-1)bni_37]y[6] + [(-1)bni_37]x[6] >= 0 & [(-1)bso_38] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (y[6] + [-1] + [-1]x[6] >= 0 & y[6] + [-1]x[6] >= 0 ==> (U^Increasing(COND_EVAL3(&&(>(y[6], x[6]), >=(y[6], x[6])), x[6], y[6])), >=) & [(-1)bni_37 + (-1)Bound*bni_37] + [(-1)bni_37]y[6] + [(-1)bni_37]x[6] >= 0 & [(-1)bso_38] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (y[6] + [-1] + [-1]x[6] >= 0 & y[6] + [-1]x[6] >= 0 ==> (U^Increasing(COND_EVAL3(&&(>(y[6], x[6]), >=(y[6], x[6])), x[6], y[6])), >=) & [(-1)bni_37 + (-1)Bound*bni_37] + [(-1)bni_37]y[6] + [(-1)bni_37]x[6] >= 0 & [(-1)bso_38] >= 0) We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (6) (y[6] >= 0 & [1] + y[6] >= 0 ==> (U^Increasing(COND_EVAL3(&&(>(y[6], x[6]), >=(y[6], x[6])), x[6], y[6])), >=) & [(-2)bni_37 + (-1)Bound*bni_37] + [(-2)bni_37]x[6] + [(-1)bni_37]y[6] >= 0 & [(-1)bso_38] >= 0) We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (7) (y[6] >= 0 & [1] + y[6] >= 0 & x[6] >= 0 ==> (U^Increasing(COND_EVAL3(&&(>(y[6], x[6]), >=(y[6], x[6])), x[6], y[6])), >=) & [(-2)bni_37 + (-1)Bound*bni_37] + [(-2)bni_37]x[6] + [(-1)bni_37]y[6] >= 0 & [(-1)bso_38] >= 0) (8) (y[6] >= 0 & [1] + y[6] >= 0 & x[6] >= 0 ==> (U^Increasing(COND_EVAL3(&&(>(y[6], x[6]), >=(y[6], x[6])), x[6], y[6])), >=) & [(-2)bni_37 + (-1)Bound*bni_37] + [(2)bni_37]x[6] + [(-1)bni_37]y[6] >= 0 & [(-1)bso_38] >= 0) For Pair COND_EVAL3(TRUE, x, y) -> EVAL(+(x, 1), y) the following chains were created: *We consider the chain EVAL(x[6], y[6]) -> COND_EVAL3(&&(>(y[6], x[6]), >=(y[6], x[6])), x[6], y[6]), COND_EVAL3(TRUE, x[7], y[7]) -> EVAL(+(x[7], 1), y[7]) which results in the following constraint: (1) (&&(>(y[6], x[6]), >=(y[6], x[6]))=TRUE & x[6]=x[7] & y[6]=y[7] ==> COND_EVAL3(TRUE, x[7], y[7])_>=_NonInfC & COND_EVAL3(TRUE, x[7], y[7])_>=_EVAL(+(x[7], 1), y[7]) & (U^Increasing(EVAL(+(x[7], 1), y[7])), >=)) We simplified constraint (1) using rules (III), (IDP_BOOLEAN) which results in the following new constraint: (2) (>(y[6], x[6])=TRUE & >=(y[6], x[6])=TRUE ==> COND_EVAL3(TRUE, x[6], y[6])_>=_NonInfC & COND_EVAL3(TRUE, x[6], y[6])_>=_EVAL(+(x[6], 1), y[6]) & (U^Increasing(EVAL(+(x[7], 1), y[7])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (y[6] + [-1] + [-1]x[6] >= 0 & y[6] + [-1]x[6] >= 0 ==> (U^Increasing(EVAL(+(x[7], 1), y[7])), >=) & [(-1)bni_39 + (-1)Bound*bni_39] + [(-1)bni_39]y[6] + [(-1)bni_39]x[6] >= 0 & [1 + (-1)bso_40] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (y[6] + [-1] + [-1]x[6] >= 0 & y[6] + [-1]x[6] >= 0 ==> (U^Increasing(EVAL(+(x[7], 1), y[7])), >=) & [(-1)bni_39 + (-1)Bound*bni_39] + [(-1)bni_39]y[6] + [(-1)bni_39]x[6] >= 0 & [1 + (-1)bso_40] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (y[6] + [-1] + [-1]x[6] >= 0 & y[6] + [-1]x[6] >= 0 ==> (U^Increasing(EVAL(+(x[7], 1), y[7])), >=) & [(-1)bni_39 + (-1)Bound*bni_39] + [(-1)bni_39]y[6] + [(-1)bni_39]x[6] >= 0 & [1 + (-1)bso_40] >= 0) We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (6) (y[6] >= 0 & [1] + y[6] >= 0 ==> (U^Increasing(EVAL(+(x[7], 1), y[7])), >=) & [(-2)bni_39 + (-1)Bound*bni_39] + [(-2)bni_39]x[6] + [(-1)bni_39]y[6] >= 0 & [1 + (-1)bso_40] >= 0) We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (7) (y[6] >= 0 & [1] + y[6] >= 0 & x[6] >= 0 ==> (U^Increasing(EVAL(+(x[7], 1), y[7])), >=) & [(-2)bni_39 + (-1)Bound*bni_39] + [(2)bni_39]x[6] + [(-1)bni_39]y[6] >= 0 & [1 + (-1)bso_40] >= 0) (8) (y[6] >= 0 & [1] + y[6] >= 0 & x[6] >= 0 ==> (U^Increasing(EVAL(+(x[7], 1), y[7])), >=) & [(-2)bni_39 + (-1)Bound*bni_39] + [(-2)bni_39]x[6] + [(-1)bni_39]y[6] >= 0 & [1 + (-1)bso_40] >= 0) To summarize, we get the following constraints P__>=_ for the following pairs. *EVAL(x, y) -> COND_EVAL(>(x, y), x, y) *(x[0] >= 0 & y[0] >= 0 ==> (U^Increasing(COND_EVAL(>(x[0], y[0]), x[0], y[0])), >=) & [(-2)bni_25 + (-1)Bound*bni_25] + [(2)bni_25]y[0] + [(-1)bni_25]x[0] >= 0 & [(-1)bso_26] >= 0) *(x[0] >= 0 & y[0] >= 0 ==> (U^Increasing(COND_EVAL(>(x[0], y[0]), x[0], y[0])), >=) & [(-2)bni_25 + (-1)Bound*bni_25] + [(-2)bni_25]y[0] + [(-1)bni_25]x[0] >= 0 & [(-1)bso_26] >= 0) *COND_EVAL(TRUE, x, y) -> EVAL(x, +(y, 1)) *(x[0] >= 0 & y[0] >= 0 ==> (U^Increasing(EVAL(x[1], +(y[1], 1))), >=) & [(-2)bni_27 + (-1)Bound*bni_27] + [(-2)bni_27]y[0] + [(-1)bni_27]x[0] >= 0 & [1 + (-1)bso_28] >= 0) *(x[0] >= 0 & y[0] >= 0 ==> (U^Increasing(EVAL(x[1], +(y[1], 1))), >=) & [(-2)bni_27 + (-1)Bound*bni_27] + [(2)bni_27]y[0] + [(-1)bni_27]x[0] >= 0 & [1 + (-1)bso_28] >= 0) *EVAL(x, y) -> COND_EVAL1(&&(>(y, x), >(x, y)), x, y) *COND_EVAL1(TRUE, x, y) -> EVAL(x, +(y, 1)) *EVAL(x, y) -> COND_EVAL2(&&(>(x, y), >=(y, x)), x, y) *COND_EVAL2(TRUE, x, y) -> EVAL(+(x, 1), y) *EVAL(x, y) -> COND_EVAL3(&&(>(y, x), >=(y, x)), x, y) *(y[6] >= 0 & [1] + y[6] >= 0 & x[6] >= 0 ==> (U^Increasing(COND_EVAL3(&&(>(y[6], x[6]), >=(y[6], x[6])), x[6], y[6])), >=) & [(-2)bni_37 + (-1)Bound*bni_37] + [(-2)bni_37]x[6] + [(-1)bni_37]y[6] >= 0 & [(-1)bso_38] >= 0) *(y[6] >= 0 & [1] + y[6] >= 0 & x[6] >= 0 ==> (U^Increasing(COND_EVAL3(&&(>(y[6], x[6]), >=(y[6], x[6])), x[6], y[6])), >=) & [(-2)bni_37 + (-1)Bound*bni_37] + [(2)bni_37]x[6] + [(-1)bni_37]y[6] >= 0 & [(-1)bso_38] >= 0) *COND_EVAL3(TRUE, x, y) -> EVAL(+(x, 1), y) *(y[6] >= 0 & [1] + y[6] >= 0 & x[6] >= 0 ==> (U^Increasing(EVAL(+(x[7], 1), y[7])), >=) & [(-2)bni_39 + (-1)Bound*bni_39] + [(2)bni_39]x[6] + [(-1)bni_39]y[6] >= 0 & [1 + (-1)bso_40] >= 0) *(y[6] >= 0 & [1] + y[6] >= 0 & x[6] >= 0 ==> (U^Increasing(EVAL(+(x[7], 1), y[7])), >=) & [(-2)bni_39 + (-1)Bound*bni_39] + [(-2)bni_39]x[6] + [(-1)bni_39]y[6] >= 0 & [1 + (-1)bso_40] >= 0) The constraints for P_> respective P_bound are constructed from P__>=_ where we just replace every occurence of "t _>=_ s" in P__>=_ by "t > s" respective "t _>=_ c". Here c stands for the fresh constant used for P_bound. Using the following integer polynomial ordering the resulting constraints can be solved Polynomial interpretation over integers[POLO]: POL(TRUE) = 0 POL(FALSE) = [1] POL(EVAL(x_1, x_2)) = [-1] + [-1]x_2 + [-1]x_1 POL(COND_EVAL(x_1, x_2, x_3)) = [-1] + [-1]x_3 + [-1]x_2 POL(>(x_1, x_2)) = [-1] POL(+(x_1, x_2)) = x_1 + x_2 POL(1) = [1] POL(COND_EVAL1(x_1, x_2, x_3)) = [-1] + [-1]x_3 + [-1]x_2 + [-1]x_1 POL(&&(x_1, x_2)) = 0 POL(COND_EVAL2(x_1, x_2, x_3)) = [-1] + [-1]x_3 + [-1]x_2 + [-1]x_1 POL(>=(x_1, x_2)) = [-1] POL(COND_EVAL3(x_1, x_2, x_3)) = [-1] + [-1]x_3 + [-1]x_2 + [-1]x_1 The following pairs are in P_>: COND_EVAL(TRUE, x[1], y[1]) -> EVAL(x[1], +(y[1], 1)) EVAL(x[2], y[2]) -> COND_EVAL1(&&(>(y[2], x[2]), >(x[2], y[2])), x[2], y[2]) COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(x[3], +(y[3], 1)) EVAL(x[4], y[4]) -> COND_EVAL2(&&(>(x[4], y[4]), >=(y[4], x[4])), x[4], y[4]) COND_EVAL2(TRUE, x[5], y[5]) -> EVAL(+(x[5], 1), y[5]) COND_EVAL3(TRUE, x[7], y[7]) -> EVAL(+(x[7], 1), y[7]) The following pairs are in P_bound: EVAL(x[2], y[2]) -> COND_EVAL1(&&(>(y[2], x[2]), >(x[2], y[2])), x[2], y[2]) COND_EVAL1(TRUE, x[3], y[3]) -> EVAL(x[3], +(y[3], 1)) EVAL(x[4], y[4]) -> COND_EVAL2(&&(>(x[4], y[4]), >=(y[4], x[4])), x[4], y[4]) COND_EVAL2(TRUE, x[5], y[5]) -> EVAL(+(x[5], 1), y[5]) The following pairs are in P_>=: EVAL(x[0], y[0]) -> COND_EVAL(>(x[0], y[0]), x[0], y[0]) EVAL(x[6], y[6]) -> COND_EVAL3(&&(>(y[6], x[6]), >=(y[6], x[6])), x[6], y[6]) At least the following rules have been oriented under context sensitive arithmetic replacement: &&(TRUE, TRUE)^1 <-> TRUE^1 FALSE^1 -> &&(TRUE, FALSE)^1 FALSE^1 -> &&(FALSE, TRUE)^1 FALSE^1 -> &&(FALSE, FALSE)^1 ---------------------------------------- (6) Obligation: IDP problem: The following function symbols are pre-defined: <<< & ~ Bwand: (Integer, Integer) -> Integer >= ~ Ge: (Integer, Integer) -> Boolean | ~ Bwor: (Integer, Integer) -> Integer / ~ Div: (Integer, Integer) -> Integer != ~ Neq: (Integer, Integer) -> Boolean && ~ Land: (Boolean, Boolean) -> Boolean ! ~ Lnot: (Boolean) -> Boolean = ~ Eq: (Integer, Integer) -> Boolean <= ~ Le: (Integer, Integer) -> Boolean ^ ~ Bwxor: (Integer, Integer) -> Integer % ~ Mod: (Integer, Integer) -> Integer + ~ Add: (Integer, Integer) -> Integer > ~ Gt: (Integer, Integer) -> Boolean -1 ~ UnaryMinus: (Integer) -> Integer < ~ Lt: (Integer, Integer) -> Boolean || ~ Lor: (Boolean, Boolean) -> Boolean - ~ Sub: (Integer, Integer) -> Integer ~ ~ Bwnot: (Integer) -> Integer * ~ Mul: (Integer, Integer) -> Integer >>> The following domains are used: Integer, Boolean R is empty. The integer pair graph contains the following rules and edges: (0): EVAL(x[0], y[0]) -> COND_EVAL(x[0] > y[0], x[0], y[0]) (1): COND_EVAL(TRUE, x[1], y[1]) -> EVAL(x[1], y[1] + 1) (6): EVAL(x[6], y[6]) -> COND_EVAL3(y[6] > x[6] && y[6] >= x[6], x[6], y[6]) (7): COND_EVAL3(TRUE, x[7], y[7]) -> EVAL(x[7] + 1, y[7]) (1) -> (0), if (x[1] ->^* x[0] & y[1] + 1 ->^* y[0]) (7) -> (0), if (x[7] + 1 ->^* x[0] & y[7] ->^* y[0]) (0) -> (1), if (x[0] > y[0] & x[0] ->^* x[1] & y[0] ->^* y[1]) (1) -> (6), if (x[1] ->^* x[6] & y[1] + 1 ->^* y[6]) (7) -> (6), if (x[7] + 1 ->^* x[6] & y[7] ->^* y[6]) (6) -> (7), if (y[6] > x[6] && y[6] >= x[6] & x[6] ->^* x[7] & y[6] ->^* y[7]) The set Q consists of the following terms: eval(x0, x1) Cond_eval(TRUE, x0, x1) Cond_eval1(TRUE, x0, x1) Cond_eval2(TRUE, x0, x1) Cond_eval3(TRUE, x0, x1) ---------------------------------------- (7) IDPNonInfProof (SOUND) Used the following options for this NonInfProof: IDPGPoloSolver: Range: [(-1,2)] IsNat: false Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpCand1ShapeHeuristic@74ba47ae Constraint Generator: NonInfConstraintGenerator: PathGenerator: MetricPathGenerator: Max Left Steps: 2 Max Right Steps: 0 The constraints were generated the following way: The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps: Note that final constraints are written in bold face. For Pair EVAL(x[0], y[0]) -> COND_EVAL(>(x[0], y[0]), x[0], y[0]) the following chains were created: *We consider the chain EVAL(x[0], y[0]) -> COND_EVAL(>(x[0], y[0]), x[0], y[0]), COND_EVAL(TRUE, x[1], y[1]) -> EVAL(x[1], +(y[1], 1)) which results in the following constraint: (1) (>(x[0], y[0])=TRUE & x[0]=x[1] & y[0]=y[1] ==> EVAL(x[0], y[0])_>=_NonInfC & EVAL(x[0], y[0])_>=_COND_EVAL(>(x[0], y[0]), x[0], y[0]) & (U^Increasing(COND_EVAL(>(x[0], y[0]), x[0], y[0])), >=)) We simplified constraint (1) using rule (IV) which results in the following new constraint: (2) (>(x[0], y[0])=TRUE ==> EVAL(x[0], y[0])_>=_NonInfC & EVAL(x[0], y[0])_>=_COND_EVAL(>(x[0], y[0]), x[0], y[0]) & (U^Increasing(COND_EVAL(>(x[0], y[0]), x[0], y[0])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (x[0] + [-1] + [-1]y[0] >= 0 ==> (U^Increasing(COND_EVAL(>(x[0], y[0]), x[0], y[0])), >=) & [(-1)bni_14 + (-1)Bound*bni_14] + [bni_14]max{y[0] + [-1]x[0], [-1]y[0] + x[0]} >= 0 & [(-1)bso_15] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (x[0] + [-1] + [-1]y[0] >= 0 ==> (U^Increasing(COND_EVAL(>(x[0], y[0]), x[0], y[0])), >=) & [(-1)bni_14 + (-1)Bound*bni_14] + [bni_14]max{y[0] + [-1]x[0], [-1]y[0] + x[0]} >= 0 & [(-1)bso_15] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (x[0] + [-1] + [-1]y[0] >= 0 & [-2]y[0] + [-1] + [2]x[0] >= 0 ==> (U^Increasing(COND_EVAL(>(x[0], y[0]), x[0], y[0])), >=) & [(-1)bni_14 + (-1)Bound*bni_14] + [(-1)bni_14]y[0] + [bni_14]x[0] >= 0 & [(-1)bso_15] >= 0) We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (6) (x[0] >= 0 & [1] + [2]x[0] >= 0 ==> (U^Increasing(COND_EVAL(>(x[0], y[0]), x[0], y[0])), >=) & [(-1)Bound*bni_14] + [bni_14]x[0] >= 0 & [(-1)bso_15] >= 0) We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (7) (x[0] >= 0 & [1] + [2]x[0] >= 0 & y[0] >= 0 ==> (U^Increasing(COND_EVAL(>(x[0], y[0]), x[0], y[0])), >=) & [(-1)Bound*bni_14] + [bni_14]x[0] >= 0 & [(-1)bso_15] >= 0) (8) (x[0] >= 0 & [1] + [2]x[0] >= 0 & y[0] >= 0 ==> (U^Increasing(COND_EVAL(>(x[0], y[0]), x[0], y[0])), >=) & [(-1)Bound*bni_14] + [bni_14]x[0] >= 0 & [(-1)bso_15] >= 0) For Pair COND_EVAL(TRUE, x[1], y[1]) -> EVAL(x[1], +(y[1], 1)) the following chains were created: *We consider the chain COND_EVAL(TRUE, x[1], y[1]) -> EVAL(x[1], +(y[1], 1)), EVAL(x[0], y[0]) -> COND_EVAL(>(x[0], y[0]), x[0], y[0]), COND_EVAL(TRUE, x[1], y[1]) -> EVAL(x[1], +(y[1], 1)) which results in the following constraint: (1) (x[1]=x[0] & +(y[1], 1)=y[0] & >(x[0], y[0])=TRUE & x[0]=x[1]1 & y[0]=y[1]1 ==> COND_EVAL(TRUE, x[1]1, y[1]1)_>=_NonInfC & COND_EVAL(TRUE, x[1]1, y[1]1)_>=_EVAL(x[1]1, +(y[1]1, 1)) & (U^Increasing(EVAL(x[1]1, +(y[1]1, 1))), >=)) We simplified constraint (1) using rule (III) which results in the following new constraint: (2) (>(x[0], +(y[1], 1))=TRUE ==> COND_EVAL(TRUE, x[0], +(y[1], 1))_>=_NonInfC & COND_EVAL(TRUE, x[0], +(y[1], 1))_>=_EVAL(x[0], +(+(y[1], 1), 1)) & (U^Increasing(EVAL(x[1]1, +(y[1]1, 1))), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (x[0] + [-2] + [-1]y[1] >= 0 ==> (U^Increasing(EVAL(x[1]1, +(y[1]1, 1))), >=) & [(-1)bni_16 + (-1)Bound*bni_16] + [bni_16]max{y[1] + [1] + [-1]x[0], [-1]y[1] + [-1] + x[0]} >= 0 & [(-1)bso_17] + max{y[1] + [1] + [-1]x[0], [-1]y[1] + [-1] + x[0]} + [-1]max{y[1] + [2] + [-1]x[0], [-1]y[1] + [-2] + x[0]} >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (x[0] + [-2] + [-1]y[1] >= 0 ==> (U^Increasing(EVAL(x[1]1, +(y[1]1, 1))), >=) & [(-1)bni_16 + (-1)Bound*bni_16] + [bni_16]max{y[1] + [1] + [-1]x[0], [-1]y[1] + [-1] + x[0]} >= 0 & [(-1)bso_17] + max{y[1] + [1] + [-1]x[0], [-1]y[1] + [-1] + x[0]} + [-1]max{y[1] + [2] + [-1]x[0], [-1]y[1] + [-2] + x[0]} >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraints: (5) (x[0] + [-2] + [-1]y[1] >= 0 & [-2]y[1] + [-3] + [2]x[0] >= 0 & [2]y[1] + [4] + [-2]x[0] >= 0 ==> (U^Increasing(EVAL(x[1]1, +(y[1]1, 1))), >=) & [(-2)bni_16 + (-1)Bound*bni_16] + [(-1)bni_16]y[1] + [bni_16]x[0] >= 0 & [-3 + (-1)bso_17] + [-2]y[1] + [2]x[0] >= 0) (6) (x[0] + [-2] + [-1]y[1] >= 0 & [-2]y[1] + [-3] + [2]x[0] >= 0 & [-2]y[1] + [-5] + [2]x[0] >= 0 ==> (U^Increasing(EVAL(x[1]1, +(y[1]1, 1))), >=) & [(-2)bni_16 + (-1)Bound*bni_16] + [(-1)bni_16]y[1] + [bni_16]x[0] >= 0 & [1 + (-1)bso_17] >= 0) We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (7) (x[0] >= 0 & [1] + [2]x[0] >= 0 & [-2]x[0] >= 0 ==> (U^Increasing(EVAL(x[1]1, +(y[1]1, 1))), >=) & [(-1)Bound*bni_16] + [bni_16]x[0] >= 0 & [1 + (-1)bso_17] + [2]x[0] >= 0) We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (8) (x[0] >= 0 & [1] + [2]x[0] >= 0 & [-1] + [2]x[0] >= 0 ==> (U^Increasing(EVAL(x[1]1, +(y[1]1, 1))), >=) & [(-1)Bound*bni_16] + [bni_16]x[0] >= 0 & [1 + (-1)bso_17] >= 0) We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (9) (0 >= 0 & [1] >= 0 & 0 >= 0 ==> (U^Increasing(EVAL(x[1]1, +(y[1]1, 1))), >=) & [(-1)Bound*bni_16] >= 0 & [1 + (-1)bso_17] >= 0) We simplified constraint (9) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (10) (0 >= 0 & [1] >= 0 & 0 >= 0 & y[1] >= 0 ==> (U^Increasing(EVAL(x[1]1, +(y[1]1, 1))), >=) & [(-1)Bound*bni_16] >= 0 & [1 + (-1)bso_17] >= 0) (11) (0 >= 0 & [1] >= 0 & 0 >= 0 & y[1] >= 0 ==> (U^Increasing(EVAL(x[1]1, +(y[1]1, 1))), >=) & [(-1)Bound*bni_16] >= 0 & [1 + (-1)bso_17] >= 0) We simplified constraint (8) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (12) (x[0] >= 0 & [1] + [2]x[0] >= 0 & [-1] + [2]x[0] >= 0 & y[1] >= 0 ==> (U^Increasing(EVAL(x[1]1, +(y[1]1, 1))), >=) & [(-1)Bound*bni_16] + [bni_16]x[0] >= 0 & [1 + (-1)bso_17] >= 0) (13) (x[0] >= 0 & [1] + [2]x[0] >= 0 & [-1] + [2]x[0] >= 0 & y[1] >= 0 ==> (U^Increasing(EVAL(x[1]1, +(y[1]1, 1))), >=) & [(-1)Bound*bni_16] + [bni_16]x[0] >= 0 & [1 + (-1)bso_17] >= 0) *We consider the chain COND_EVAL3(TRUE, x[7], y[7]) -> EVAL(+(x[7], 1), y[7]), EVAL(x[0], y[0]) -> COND_EVAL(>(x[0], y[0]), x[0], y[0]), COND_EVAL(TRUE, x[1], y[1]) -> EVAL(x[1], +(y[1], 1)) which results in the following constraint: (1) (+(x[7], 1)=x[0] & y[7]=y[0] & >(x[0], y[0])=TRUE & x[0]=x[1] & y[0]=y[1] ==> COND_EVAL(TRUE, x[1], y[1])_>=_NonInfC & COND_EVAL(TRUE, x[1], y[1])_>=_EVAL(x[1], +(y[1], 1)) & (U^Increasing(EVAL(x[1], +(y[1], 1))), >=)) We simplified constraint (1) using rule (III) which results in the following new constraint: (2) (>(+(x[7], 1), y[0])=TRUE ==> COND_EVAL(TRUE, +(x[7], 1), y[0])_>=_NonInfC & COND_EVAL(TRUE, +(x[7], 1), y[0])_>=_EVAL(+(x[7], 1), +(y[0], 1)) & (U^Increasing(EVAL(x[1], +(y[1], 1))), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (x[7] + [-1]y[0] >= 0 ==> (U^Increasing(EVAL(x[1], +(y[1], 1))), >=) & [(-1)bni_16 + (-1)Bound*bni_16] + [bni_16]max{y[0] + [-1] + [-1]x[7], [-1]y[0] + [1] + x[7]} >= 0 & [(-1)bso_17] + max{y[0] + [-1] + [-1]x[7], [-1]y[0] + [1] + x[7]} + [-1]max{y[0] + [-1]x[7], [-1]y[0] + x[7]} >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (x[7] + [-1]y[0] >= 0 ==> (U^Increasing(EVAL(x[1], +(y[1], 1))), >=) & [(-1)bni_16 + (-1)Bound*bni_16] + [bni_16]max{y[0] + [-1] + [-1]x[7], [-1]y[0] + [1] + x[7]} >= 0 & [(-1)bso_17] + max{y[0] + [-1] + [-1]x[7], [-1]y[0] + [1] + x[7]} + [-1]max{y[0] + [-1]x[7], [-1]y[0] + x[7]} >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraints: (5) (x[7] + [-1]y[0] >= 0 & [-2]y[0] + [1] + [2]x[7] >= 0 & [2]y[0] + [-2]x[7] >= 0 ==> (U^Increasing(EVAL(x[1], +(y[1], 1))), >=) & [(-1)Bound*bni_16] + [(-1)bni_16]y[0] + [bni_16]x[7] >= 0 & [1 + (-1)bso_17] + [-2]y[0] + [2]x[7] >= 0) (6) (x[7] + [-1]y[0] >= 0 & [-2]y[0] + [1] + [2]x[7] >= 0 & [-2]y[0] + [-1] + [2]x[7] >= 0 ==> (U^Increasing(EVAL(x[1], +(y[1], 1))), >=) & [(-1)Bound*bni_16] + [(-1)bni_16]y[0] + [bni_16]x[7] >= 0 & [1 + (-1)bso_17] >= 0) We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (7) (x[7] >= 0 & [1] + [2]x[7] >= 0 & [-2]x[7] >= 0 ==> (U^Increasing(EVAL(x[1], +(y[1], 1))), >=) & [(-1)Bound*bni_16] + [bni_16]x[7] >= 0 & [1 + (-1)bso_17] + [2]x[7] >= 0) We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (8) (x[7] >= 0 & [1] + [2]x[7] >= 0 & [-1] + [2]x[7] >= 0 ==> (U^Increasing(EVAL(x[1], +(y[1], 1))), >=) & [(-1)Bound*bni_16] + [bni_16]x[7] >= 0 & [1 + (-1)bso_17] >= 0) We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (9) (0 >= 0 & [1] >= 0 & 0 >= 0 ==> (U^Increasing(EVAL(x[1], +(y[1], 1))), >=) & [(-1)Bound*bni_16] >= 0 & [1 + (-1)bso_17] >= 0) We simplified constraint (9) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (10) (0 >= 0 & [1] >= 0 & 0 >= 0 & y[0] >= 0 ==> (U^Increasing(EVAL(x[1], +(y[1], 1))), >=) & [(-1)Bound*bni_16] >= 0 & [1 + (-1)bso_17] >= 0) (11) (0 >= 0 & [1] >= 0 & 0 >= 0 & y[0] >= 0 ==> (U^Increasing(EVAL(x[1], +(y[1], 1))), >=) & [(-1)Bound*bni_16] >= 0 & [1 + (-1)bso_17] >= 0) We simplified constraint (8) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (12) (x[7] >= 0 & [1] + [2]x[7] >= 0 & [-1] + [2]x[7] >= 0 & y[0] >= 0 ==> (U^Increasing(EVAL(x[1], +(y[1], 1))), >=) & [(-1)Bound*bni_16] + [bni_16]x[7] >= 0 & [1 + (-1)bso_17] >= 0) (13) (x[7] >= 0 & [1] + [2]x[7] >= 0 & [-1] + [2]x[7] >= 0 & y[0] >= 0 ==> (U^Increasing(EVAL(x[1], +(y[1], 1))), >=) & [(-1)Bound*bni_16] + [bni_16]x[7] >= 0 & [1 + (-1)bso_17] >= 0) For Pair EVAL(x[6], y[6]) -> COND_EVAL3(&&(>(y[6], x[6]), >=(y[6], x[6])), x[6], y[6]) the following chains were created: *We consider the chain EVAL(x[6], y[6]) -> COND_EVAL3(&&(>(y[6], x[6]), >=(y[6], x[6])), x[6], y[6]), COND_EVAL3(TRUE, x[7], y[7]) -> EVAL(+(x[7], 1), y[7]) which results in the following constraint: (1) (&&(>(y[6], x[6]), >=(y[6], x[6]))=TRUE & x[6]=x[7] & y[6]=y[7] ==> EVAL(x[6], y[6])_>=_NonInfC & EVAL(x[6], y[6])_>=_COND_EVAL3(&&(>(y[6], x[6]), >=(y[6], x[6])), x[6], y[6]) & (U^Increasing(COND_EVAL3(&&(>(y[6], x[6]), >=(y[6], x[6])), x[6], y[6])), >=)) We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint: (2) (>(y[6], x[6])=TRUE & >=(y[6], x[6])=TRUE ==> EVAL(x[6], y[6])_>=_NonInfC & EVAL(x[6], y[6])_>=_COND_EVAL3(&&(>(y[6], x[6]), >=(y[6], x[6])), x[6], y[6]) & (U^Increasing(COND_EVAL3(&&(>(y[6], x[6]), >=(y[6], x[6])), x[6], y[6])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (y[6] + [-1] + [-1]x[6] >= 0 & y[6] + [-1]x[6] >= 0 ==> (U^Increasing(COND_EVAL3(&&(>(y[6], x[6]), >=(y[6], x[6])), x[6], y[6])), >=) & [(-1)bni_18 + (-1)Bound*bni_18] + [bni_18]max{y[6] + [-1]x[6], [-1]y[6] + x[6]} >= 0 & [(-1)bso_19] >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (y[6] + [-1] + [-1]x[6] >= 0 & y[6] + [-1]x[6] >= 0 ==> (U^Increasing(COND_EVAL3(&&(>(y[6], x[6]), >=(y[6], x[6])), x[6], y[6])), >=) & [(-1)bni_18 + (-1)Bound*bni_18] + [bni_18]max{y[6] + [-1]x[6], [-1]y[6] + x[6]} >= 0 & [(-1)bso_19] >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (y[6] + [-1] + [-1]x[6] >= 0 & y[6] + [-1]x[6] >= 0 & [2]y[6] + [-2]x[6] >= 0 ==> (U^Increasing(COND_EVAL3(&&(>(y[6], x[6]), >=(y[6], x[6])), x[6], y[6])), >=) & [(-1)bni_18 + (-1)Bound*bni_18] + [bni_18]y[6] + [(-1)bni_18]x[6] >= 0 & [(-1)bso_19] >= 0) We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (6) (y[6] >= 0 & [1] + y[6] >= 0 & [2] + [2]y[6] >= 0 ==> (U^Increasing(COND_EVAL3(&&(>(y[6], x[6]), >=(y[6], x[6])), x[6], y[6])), >=) & [(-1)Bound*bni_18] + [bni_18]y[6] >= 0 & [(-1)bso_19] >= 0) We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (7) (y[6] >= 0 & [1] + y[6] >= 0 & [2] + [2]y[6] >= 0 & x[6] >= 0 ==> (U^Increasing(COND_EVAL3(&&(>(y[6], x[6]), >=(y[6], x[6])), x[6], y[6])), >=) & [(-1)Bound*bni_18] + [bni_18]y[6] >= 0 & [(-1)bso_19] >= 0) (8) (y[6] >= 0 & [1] + y[6] >= 0 & [2] + [2]y[6] >= 0 & x[6] >= 0 ==> (U^Increasing(COND_EVAL3(&&(>(y[6], x[6]), >=(y[6], x[6])), x[6], y[6])), >=) & [(-1)Bound*bni_18] + [bni_18]y[6] >= 0 & [(-1)bso_19] >= 0) We simplified constraint (7) using rule (IDP_POLY_GCD) which results in the following new constraint: (9) (y[6] >= 0 & [1] + y[6] >= 0 & x[6] >= 0 & [1] + y[6] >= 0 ==> (U^Increasing(COND_EVAL3(&&(>(y[6], x[6]), >=(y[6], x[6])), x[6], y[6])), >=) & [(-1)Bound*bni_18] + [bni_18]y[6] >= 0 & [(-1)bso_19] >= 0) We simplified constraint (8) using rule (IDP_POLY_GCD) which results in the following new constraint: (10) (y[6] >= 0 & [1] + y[6] >= 0 & x[6] >= 0 & [1] + y[6] >= 0 ==> (U^Increasing(COND_EVAL3(&&(>(y[6], x[6]), >=(y[6], x[6])), x[6], y[6])), >=) & [(-1)Bound*bni_18] + [bni_18]y[6] >= 0 & [(-1)bso_19] >= 0) For Pair COND_EVAL3(TRUE, x[7], y[7]) -> EVAL(+(x[7], 1), y[7]) the following chains were created: *We consider the chain COND_EVAL(TRUE, x[1], y[1]) -> EVAL(x[1], +(y[1], 1)), EVAL(x[6], y[6]) -> COND_EVAL3(&&(>(y[6], x[6]), >=(y[6], x[6])), x[6], y[6]), COND_EVAL3(TRUE, x[7], y[7]) -> EVAL(+(x[7], 1), y[7]) which results in the following constraint: (1) (x[1]=x[6] & +(y[1], 1)=y[6] & &&(>(y[6], x[6]), >=(y[6], x[6]))=TRUE & x[6]=x[7] & y[6]=y[7] ==> COND_EVAL3(TRUE, x[7], y[7])_>=_NonInfC & COND_EVAL3(TRUE, x[7], y[7])_>=_EVAL(+(x[7], 1), y[7]) & (U^Increasing(EVAL(+(x[7], 1), y[7])), >=)) We simplified constraint (1) using rules (III), (IDP_BOOLEAN) which results in the following new constraint: (2) (>(+(y[1], 1), x[6])=TRUE & >=(+(y[1], 1), x[6])=TRUE ==> COND_EVAL3(TRUE, x[6], +(y[1], 1))_>=_NonInfC & COND_EVAL3(TRUE, x[6], +(y[1], 1))_>=_EVAL(+(x[6], 1), +(y[1], 1)) & (U^Increasing(EVAL(+(x[7], 1), y[7])), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (y[1] + [-1]x[6] >= 0 & y[1] + [1] + [-1]x[6] >= 0 ==> (U^Increasing(EVAL(+(x[7], 1), y[7])), >=) & [(-1)bni_20 + (-1)Bound*bni_20] + [bni_20]max{y[1] + [1] + [-1]x[6], [-1]y[1] + [-1] + x[6]} >= 0 & [(-1)bso_21] + max{y[1] + [1] + [-1]x[6], [-1]y[1] + [-1] + x[6]} + [-1]max{y[1] + [-1]x[6], [-1]y[1] + x[6]} >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (y[1] + [-1]x[6] >= 0 & y[1] + [1] + [-1]x[6] >= 0 ==> (U^Increasing(EVAL(+(x[7], 1), y[7])), >=) & [(-1)bni_20 + (-1)Bound*bni_20] + [bni_20]max{y[1] + [1] + [-1]x[6], [-1]y[1] + [-1] + x[6]} >= 0 & [(-1)bso_21] + max{y[1] + [1] + [-1]x[6], [-1]y[1] + [-1] + x[6]} + [-1]max{y[1] + [-1]x[6], [-1]y[1] + x[6]} >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (y[1] + [-1]x[6] >= 0 & y[1] + [1] + [-1]x[6] >= 0 & [2]y[1] + [2] + [-2]x[6] >= 0 & [2]y[1] + [-2]x[6] >= 0 ==> (U^Increasing(EVAL(+(x[7], 1), y[7])), >=) & [(-1)Bound*bni_20] + [bni_20]y[1] + [(-1)bni_20]x[6] >= 0 & [1 + (-1)bso_21] >= 0) We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (6) (y[1] >= 0 & [1] + y[1] >= 0 & [2] + [2]y[1] >= 0 & [2]y[1] >= 0 ==> (U^Increasing(EVAL(+(x[7], 1), y[7])), >=) & [(-1)Bound*bni_20] + [bni_20]y[1] >= 0 & [1 + (-1)bso_21] >= 0) We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (7) (y[1] >= 0 & [1] + y[1] >= 0 & [2] + [2]y[1] >= 0 & [2]y[1] >= 0 & x[6] >= 0 ==> (U^Increasing(EVAL(+(x[7], 1), y[7])), >=) & [(-1)Bound*bni_20] + [bni_20]y[1] >= 0 & [1 + (-1)bso_21] >= 0) (8) (y[1] >= 0 & [1] + y[1] >= 0 & [2] + [2]y[1] >= 0 & [2]y[1] >= 0 & x[6] >= 0 ==> (U^Increasing(EVAL(+(x[7], 1), y[7])), >=) & [(-1)Bound*bni_20] + [bni_20]y[1] >= 0 & [1 + (-1)bso_21] >= 0) We simplified constraint (7) using rule (IDP_POLY_GCD) which results in the following new constraint: (9) (y[1] >= 0 & [1] + y[1] >= 0 & x[6] >= 0 & [1] + y[1] >= 0 & y[1] >= 0 ==> (U^Increasing(EVAL(+(x[7], 1), y[7])), >=) & [(-1)Bound*bni_20] + [bni_20]y[1] >= 0 & [1 + (-1)bso_21] >= 0) We simplified constraint (8) using rule (IDP_POLY_GCD) which results in the following new constraint: (10) (y[1] >= 0 & [1] + y[1] >= 0 & x[6] >= 0 & [1] + y[1] >= 0 & y[1] >= 0 ==> (U^Increasing(EVAL(+(x[7], 1), y[7])), >=) & [(-1)Bound*bni_20] + [bni_20]y[1] >= 0 & [1 + (-1)bso_21] >= 0) *We consider the chain COND_EVAL3(TRUE, x[7], y[7]) -> EVAL(+(x[7], 1), y[7]), EVAL(x[6], y[6]) -> COND_EVAL3(&&(>(y[6], x[6]), >=(y[6], x[6])), x[6], y[6]), COND_EVAL3(TRUE, x[7], y[7]) -> EVAL(+(x[7], 1), y[7]) which results in the following constraint: (1) (+(x[7], 1)=x[6] & y[7]=y[6] & &&(>(y[6], x[6]), >=(y[6], x[6]))=TRUE & x[6]=x[7]1 & y[6]=y[7]1 ==> COND_EVAL3(TRUE, x[7]1, y[7]1)_>=_NonInfC & COND_EVAL3(TRUE, x[7]1, y[7]1)_>=_EVAL(+(x[7]1, 1), y[7]1) & (U^Increasing(EVAL(+(x[7]1, 1), y[7]1)), >=)) We simplified constraint (1) using rules (III), (IDP_BOOLEAN) which results in the following new constraint: (2) (>(y[6], +(x[7], 1))=TRUE & >=(y[6], +(x[7], 1))=TRUE ==> COND_EVAL3(TRUE, +(x[7], 1), y[6])_>=_NonInfC & COND_EVAL3(TRUE, +(x[7], 1), y[6])_>=_EVAL(+(+(x[7], 1), 1), y[6]) & (U^Increasing(EVAL(+(x[7]1, 1), y[7]1)), >=)) We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint: (3) (y[6] + [-2] + [-1]x[7] >= 0 & y[6] + [-1] + [-1]x[7] >= 0 ==> (U^Increasing(EVAL(+(x[7]1, 1), y[7]1)), >=) & [(-1)bni_20 + (-1)Bound*bni_20] + [bni_20]max{y[6] + [-1] + [-1]x[7], [-1]y[6] + [1] + x[7]} >= 0 & [(-1)bso_21] + max{y[6] + [-1] + [-1]x[7], [-1]y[6] + [1] + x[7]} + [-1]max{y[6] + [-2] + [-1]x[7], [-1]y[6] + [2] + x[7]} >= 0) We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint: (4) (y[6] + [-2] + [-1]x[7] >= 0 & y[6] + [-1] + [-1]x[7] >= 0 ==> (U^Increasing(EVAL(+(x[7]1, 1), y[7]1)), >=) & [(-1)bni_20 + (-1)Bound*bni_20] + [bni_20]max{y[6] + [-1] + [-1]x[7], [-1]y[6] + [1] + x[7]} >= 0 & [(-1)bso_21] + max{y[6] + [-1] + [-1]x[7], [-1]y[6] + [1] + x[7]} + [-1]max{y[6] + [-2] + [-1]x[7], [-1]y[6] + [2] + x[7]} >= 0) We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint: (5) (y[6] + [-2] + [-1]x[7] >= 0 & y[6] + [-1] + [-1]x[7] >= 0 & [2]y[6] + [-2] + [-2]x[7] >= 0 & [2]y[6] + [-4] + [-2]x[7] >= 0 ==> (U^Increasing(EVAL(+(x[7]1, 1), y[7]1)), >=) & [(-2)bni_20 + (-1)Bound*bni_20] + [bni_20]y[6] + [(-1)bni_20]x[7] >= 0 & [1 + (-1)bso_21] >= 0) We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint: (6) (y[6] >= 0 & [1] + y[6] >= 0 & [2] + [2]y[6] >= 0 & [2]y[6] >= 0 ==> (U^Increasing(EVAL(+(x[7]1, 1), y[7]1)), >=) & [(-1)Bound*bni_20] + [bni_20]y[6] >= 0 & [1 + (-1)bso_21] >= 0) We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraints: (7) (y[6] >= 0 & [1] + y[6] >= 0 & [2] + [2]y[6] >= 0 & [2]y[6] >= 0 & x[7] >= 0 ==> (U^Increasing(EVAL(+(x[7]1, 1), y[7]1)), >=) & [(-1)Bound*bni_20] + [bni_20]y[6] >= 0 & [1 + (-1)bso_21] >= 0) (8) (y[6] >= 0 & [1] + y[6] >= 0 & [2] + [2]y[6] >= 0 & [2]y[6] >= 0 & x[7] >= 0 ==> (U^Increasing(EVAL(+(x[7]1, 1), y[7]1)), >=) & [(-1)Bound*bni_20] + [bni_20]y[6] >= 0 & [1 + (-1)bso_21] >= 0) We simplified constraint (7) using rule (IDP_POLY_GCD) which results in the following new constraint: (9) (y[6] >= 0 & [1] + y[6] >= 0 & x[7] >= 0 & [1] + y[6] >= 0 & y[6] >= 0 ==> (U^Increasing(EVAL(+(x[7]1, 1), y[7]1)), >=) & [(-1)Bound*bni_20] + [bni_20]y[6] >= 0 & [1 + (-1)bso_21] >= 0) We simplified constraint (8) using rule (IDP_POLY_GCD) which results in the following new constraint: (10) (y[6] >= 0 & [1] + y[6] >= 0 & x[7] >= 0 & [1] + y[6] >= 0 & y[6] >= 0 ==> (U^Increasing(EVAL(+(x[7]1, 1), y[7]1)), >=) & [(-1)Bound*bni_20] + [bni_20]y[6] >= 0 & [1 + (-1)bso_21] >= 0) To summarize, we get the following constraints P__>=_ for the following pairs. *EVAL(x[0], y[0]) -> COND_EVAL(>(x[0], y[0]), x[0], y[0]) *(x[0] >= 0 & [1] + [2]x[0] >= 0 & y[0] >= 0 ==> (U^Increasing(COND_EVAL(>(x[0], y[0]), x[0], y[0])), >=) & [(-1)Bound*bni_14] + [bni_14]x[0] >= 0 & [(-1)bso_15] >= 0) *(x[0] >= 0 & [1] + [2]x[0] >= 0 & y[0] >= 0 ==> (U^Increasing(COND_EVAL(>(x[0], y[0]), x[0], y[0])), >=) & [(-1)Bound*bni_14] + [bni_14]x[0] >= 0 & [(-1)bso_15] >= 0) *COND_EVAL(TRUE, x[1], y[1]) -> EVAL(x[1], +(y[1], 1)) *(0 >= 0 & [1] >= 0 & 0 >= 0 & y[1] >= 0 ==> (U^Increasing(EVAL(x[1]1, +(y[1]1, 1))), >=) & [(-1)Bound*bni_16] >= 0 & [1 + (-1)bso_17] >= 0) *(0 >= 0 & [1] >= 0 & 0 >= 0 & y[1] >= 0 ==> (U^Increasing(EVAL(x[1]1, +(y[1]1, 1))), >=) & [(-1)Bound*bni_16] >= 0 & [1 + (-1)bso_17] >= 0) *(x[0] >= 0 & [1] + [2]x[0] >= 0 & [-1] + [2]x[0] >= 0 & y[1] >= 0 ==> (U^Increasing(EVAL(x[1]1, +(y[1]1, 1))), >=) & [(-1)Bound*bni_16] + [bni_16]x[0] >= 0 & [1 + (-1)bso_17] >= 0) *(x[0] >= 0 & [1] + [2]x[0] >= 0 & [-1] + [2]x[0] >= 0 & y[1] >= 0 ==> (U^Increasing(EVAL(x[1]1, +(y[1]1, 1))), >=) & [(-1)Bound*bni_16] + [bni_16]x[0] >= 0 & [1 + (-1)bso_17] >= 0) *(0 >= 0 & [1] >= 0 & 0 >= 0 & y[0] >= 0 ==> (U^Increasing(EVAL(x[1], +(y[1], 1))), >=) & [(-1)Bound*bni_16] >= 0 & [1 + (-1)bso_17] >= 0) *(0 >= 0 & [1] >= 0 & 0 >= 0 & y[0] >= 0 ==> (U^Increasing(EVAL(x[1], +(y[1], 1))), >=) & [(-1)Bound*bni_16] >= 0 & [1 + (-1)bso_17] >= 0) *(x[7] >= 0 & [1] + [2]x[7] >= 0 & [-1] + [2]x[7] >= 0 & y[0] >= 0 ==> (U^Increasing(EVAL(x[1], +(y[1], 1))), >=) & [(-1)Bound*bni_16] + [bni_16]x[7] >= 0 & [1 + (-1)bso_17] >= 0) *(x[7] >= 0 & [1] + [2]x[7] >= 0 & [-1] + [2]x[7] >= 0 & y[0] >= 0 ==> (U^Increasing(EVAL(x[1], +(y[1], 1))), >=) & [(-1)Bound*bni_16] + [bni_16]x[7] >= 0 & [1 + (-1)bso_17] >= 0) *EVAL(x[6], y[6]) -> COND_EVAL3(&&(>(y[6], x[6]), >=(y[6], x[6])), x[6], y[6]) *(y[6] >= 0 & [1] + y[6] >= 0 & x[6] >= 0 & [1] + y[6] >= 0 ==> (U^Increasing(COND_EVAL3(&&(>(y[6], x[6]), >=(y[6], x[6])), x[6], y[6])), >=) & [(-1)Bound*bni_18] + [bni_18]y[6] >= 0 & [(-1)bso_19] >= 0) *(y[6] >= 0 & [1] + y[6] >= 0 & x[6] >= 0 & [1] + y[6] >= 0 ==> (U^Increasing(COND_EVAL3(&&(>(y[6], x[6]), >=(y[6], x[6])), x[6], y[6])), >=) & [(-1)Bound*bni_18] + [bni_18]y[6] >= 0 & [(-1)bso_19] >= 0) *COND_EVAL3(TRUE, x[7], y[7]) -> EVAL(+(x[7], 1), y[7]) *(y[1] >= 0 & [1] + y[1] >= 0 & x[6] >= 0 & [1] + y[1] >= 0 & y[1] >= 0 ==> (U^Increasing(EVAL(+(x[7], 1), y[7])), >=) & [(-1)Bound*bni_20] + [bni_20]y[1] >= 0 & [1 + (-1)bso_21] >= 0) *(y[1] >= 0 & [1] + y[1] >= 0 & x[6] >= 0 & [1] + y[1] >= 0 & y[1] >= 0 ==> (U^Increasing(EVAL(+(x[7], 1), y[7])), >=) & [(-1)Bound*bni_20] + [bni_20]y[1] >= 0 & [1 + (-1)bso_21] >= 0) *(y[6] >= 0 & [1] + y[6] >= 0 & x[7] >= 0 & [1] + y[6] >= 0 & y[6] >= 0 ==> (U^Increasing(EVAL(+(x[7]1, 1), y[7]1)), >=) & [(-1)Bound*bni_20] + [bni_20]y[6] >= 0 & [1 + (-1)bso_21] >= 0) *(y[6] >= 0 & [1] + y[6] >= 0 & x[7] >= 0 & [1] + y[6] >= 0 & y[6] >= 0 ==> (U^Increasing(EVAL(+(x[7]1, 1), y[7]1)), >=) & [(-1)Bound*bni_20] + [bni_20]y[6] >= 0 & [1 + (-1)bso_21] >= 0) The constraints for P_> respective P_bound are constructed from P__>=_ where we just replace every occurence of "t _>=_ s" in P__>=_ by "t > s" respective "t _>=_ c". Here c stands for the fresh constant used for P_bound. Using the following integer polynomial ordering the resulting constraints can be solved Polynomial interpretation over integers[POLO]: POL(TRUE) = 0 POL(FALSE) = 0 POL(EVAL(x_1, x_2)) = [-1] + max{x_2 + [-1]x_1, [-1]x_2 + x_1} POL(COND_EVAL(x_1, x_2, x_3)) = [-1] + max{x_3 + [-1]x_2, [-1]x_3 + x_2} POL(>(x_1, x_2)) = [-1] POL(+(x_1, x_2)) = x_1 + x_2 POL(1) = [1] POL(COND_EVAL3(x_1, x_2, x_3)) = [-1] + max{x_3 + [-1]x_2, [-1]x_3 + x_2} POL(&&(x_1, x_2)) = [-1] POL(>=(x_1, x_2)) = [-1] The following pairs are in P_>: COND_EVAL(TRUE, x[1], y[1]) -> EVAL(x[1], +(y[1], 1)) COND_EVAL3(TRUE, x[7], y[7]) -> EVAL(+(x[7], 1), y[7]) The following pairs are in P_bound: EVAL(x[0], y[0]) -> COND_EVAL(>(x[0], y[0]), x[0], y[0]) COND_EVAL(TRUE, x[1], y[1]) -> EVAL(x[1], +(y[1], 1)) EVAL(x[6], y[6]) -> COND_EVAL3(&&(>(y[6], x[6]), >=(y[6], x[6])), x[6], y[6]) COND_EVAL3(TRUE, x[7], y[7]) -> EVAL(+(x[7], 1), y[7]) The following pairs are in P_>=: EVAL(x[0], y[0]) -> COND_EVAL(>(x[0], y[0]), x[0], y[0]) EVAL(x[6], y[6]) -> COND_EVAL3(&&(>(y[6], x[6]), >=(y[6], x[6])), x[6], y[6]) There are no usable rules. ---------------------------------------- (8) Obligation: IDP problem: The following function symbols are pre-defined: <<< & ~ Bwand: (Integer, Integer) -> Integer >= ~ Ge: (Integer, Integer) -> Boolean | ~ Bwor: (Integer, Integer) -> Integer / ~ Div: (Integer, Integer) -> Integer != ~ Neq: (Integer, Integer) -> Boolean && ~ Land: (Boolean, Boolean) -> Boolean ! ~ Lnot: (Boolean) -> Boolean = ~ Eq: (Integer, Integer) -> Boolean <= ~ Le: (Integer, Integer) -> Boolean ^ ~ Bwxor: (Integer, Integer) -> Integer % ~ Mod: (Integer, Integer) -> Integer + ~ Add: (Integer, Integer) -> Integer > ~ Gt: (Integer, Integer) -> Boolean -1 ~ UnaryMinus: (Integer) -> Integer < ~ Lt: (Integer, Integer) -> Boolean || ~ Lor: (Boolean, Boolean) -> Boolean - ~ Sub: (Integer, Integer) -> Integer ~ ~ Bwnot: (Integer) -> Integer * ~ Mul: (Integer, Integer) -> Integer >>> The following domains are used: Integer, Boolean R is empty. The integer pair graph contains the following rules and edges: (0): EVAL(x[0], y[0]) -> COND_EVAL(x[0] > y[0], x[0], y[0]) (6): EVAL(x[6], y[6]) -> COND_EVAL3(y[6] > x[6] && y[6] >= x[6], x[6], y[6]) The set Q consists of the following terms: eval(x0, x1) Cond_eval(TRUE, x0, x1) Cond_eval1(TRUE, x0, x1) Cond_eval2(TRUE, x0, x1) Cond_eval3(TRUE, x0, x1) ---------------------------------------- (9) IDependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes. ---------------------------------------- (10) TRUE